Résumé
Incorporating softening behaviors in scales transition problems is a complex task, especially due to strain and stress localization. Micromechanical approaches (Gurson-type models) have to invoke some ad hoc coalescence models in order to reproduce relevant overall fracture [1]. Numerical simulations based on continuum damage mechanics lead to a vanishing energy (energy dissipated due to strain-softening damage) when the mesh tends to zero and nonlocal damage theories have to be formulated [2]. Recently, a renewed interest is noticed in the literature for these nonlocal (explicit, integral or variational) damage models in particular concerning the diffuse microcracking observed in quasi-brittle materials before the onset of a localized macrocrack (see [3], [4], [5] and references therein). The new micromechanical approach, presented here, allows multiscale transitions in softening problems (up to fracture) without any nonlocal or first gradient theory. This approach, first proposed by [6], is based on a cohesive-volumetric finite element method together with rigorous theoretical upper bounds on the overall strain potential. The surface damage is defined through a modified secant modulus of any intrinsic cohesive zone model (traction-separation law with initial stiffness). Considering each face of a three dimensional mesh as a penny shaped damageable inclusion, a linear comparison composite (LCC) is defined using the variational approach of [7]. A convenient estimate of the LCC properties and an asymptotic analysis lead to a micromechanical damage model based on the cohesive zone approach. This model includes a dependence to the mesh size allowing finite values of the overall properties (stiffness, dissipated energy, maximal stress, etc.) at mesh convergence. This model used in complex multiscale problems as fracture of functionally graded materials looks very promising.